Abstract: The grand canonical ensemble method is an important method in quantum statistical mechanics. The grand canonical method is generally used to deal with open systems. The grand canonical method is also used to deal with isolated systems as an approximation method. It is easier for the grand canonical method than the micro canonical ensemble method in dealing with identical particles. In exchange, there are discrepancies in the results of the grand canonical ensemble method when dealing with isolated systems.  The discrepancy of macroscopic quantities is provided when the particle number goes to infinity. The discrepancy comes from the different between the particle distribution of the grand canonical ensemble and of the micro canonical ensemble.  In this paper, we construct an isolated system which can be exactly solved with the micro canonical ensemble method. By comparing the particle distribution in the grand canonical ensemble method and in the micro canonical ensemble method, we show the discrepancy directly. The result shows that the discrepancy is larger when we have less particle number in the system. The discrepancy of ground states is larger than excited states. The discrepancy of Bose systems is larger than Fermi systems. 

Key words:  Ensemble theory, Particle distribution, Quantum statistics